June  2021, 20(6): 2323-2340. doi: 10.3934/cpaa.2021081

On local solvability for a class of generalized Mizohata equations

1. 

Universidade Federal do Rio Grande, Instituto de Matemática, Estatística e Física, RS, Brazil

2. 

Universidade Federal de São Carlos, Departamento de Matemáica, SP, Brazil

* Corresponding author

Received  October 2020 Revised  March 2021 Published  June 2021 Early access  May 2021

Fund Project: The first author was partially supported by CAPES. The second author is a associate researcher of the Thematic project no. 2018/14316-3 (FAPESP)

The image in $ C^{\infty} $ for a class of complex vector fields, containing the Mizohata operator, was characterized.

Citation: L. R. Nunes, J. R. Dos Santos Filho. On local solvability for a class of generalized Mizohata equations. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2323-2340. doi: 10.3934/cpaa.2021081
References:
[1]

M. S. BaouendiC. H. Chang and F. Treves, Microlocal Hypo-Analytic and Extension of CR Functions, J. Differ. Geom., 18 (1983), 331-391.   Google Scholar

[2]

M. S. Baouendi and F. Treves, A local constancy principle for the solutions of certain overdetermined systems of first-order linear partial differential equations, in Advances in Mathematics Supplementary Studies, Academic Press, New York, (1981), 245-262.  Google Scholar

[3]

S. Berhanu, P. D. Cordaro and J. Hounie, An introduction to involutive structures, New mathematical monographs, University Press, (2008).  Google Scholar

[4]

P. D. Cordaro, Resolubilidade das Equações Diferenciais Parciais Lineares, Matemática Universitária, 14, (1992), 51-67. Google Scholar

[5]

P. D. Cordaro, Sistemas de Campos Vetoriais Complexos, Instituto de matemática pura e aplicada, (1986). Google Scholar

[6]

L. Garding and B. Malgrange, Opérateurs différentiels partiellement hypoelliptiques et partiellement elliptiques, Math. Scand., 9 (1961), 5-21.  doi: 10.7146/math.scand.a-10619.  Google Scholar

[7]

V. Grushin, A differential Equation without a solution, Mathematical Notes, 10 (1971), 499-501.   Google Scholar

[8]

N. Hanges, Almost mizohata operators, Trans. Amer. Math. Soc., 2 (1986), 663-675.  doi: 10.2307/2000030.  Google Scholar

[9]

G. Hoepfner and R. Medrado, Microlocal regularity for Mizohata type differential operators, J. Inst. Math. Jussieu, 19 (2020), 1185-1209.  doi: 10.1017/S1474748018000361.  Google Scholar

[10]

L. Hörmander, Differential equations without solutions, Math. Ann., 140 (1960), 169-173.  doi: 10.1007/BF01361142.  Google Scholar

[11]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Springer-Verlag, 1989. doi: 10.1007/978-3-642-61497-2.  Google Scholar

[12]

H. Lewy, An example of a smooth linear partial differential equation without solution, Ann. Math., 66 (1957), 155-158.  doi: 10.2307/1970121.  Google Scholar

[13]

S. Mizohata, Une remarque sur les opérateurs différentielshypoelliptiques et partiellement hypoelliptiques, J. Math. Kyoto Univ., 1-3 (1962), 411-423.  doi: 10.1215/kjm/1250525013.  Google Scholar

[14]

H. Ninomiya, A necessary and sufficient condition of local integrability, J. Math Kyoto Univ., 39-4 (1999), 685-696.  doi: 10.1215/kjm/1250517821.  Google Scholar

[15]

L. Nirenberg and F. Treves, On local solvability of linear partial differential equations, I. Necessary conditions, Commun. Pure Appl. Math, 23 (1970), 1-38.  doi: 10.1002/cpa.3160230102.  Google Scholar

[16]

L. Nirenberg and F. Treves, On local solvability of linear partial differential equations, II. Sufficient conditions, Commun. Pure Appl. Math, 23 (1970), 459-510.  doi: 10.1002/cpa.3160230314.  Google Scholar

[17]

L. Nirenberg and F. Treves, Solvability of a order linear partial differential equation, Commun. Pure Appl. Math, 16 (1963), 331-351.  doi: 10.1002/cpa.3160160308.  Google Scholar

[18]

L. Nunes, Resolubilidade Local Para Duas Classes de Campos de Vetores Suaves Complexos, Ph. D. thesis (in portuguese), UFSCar, 2016. Google Scholar

[19]

F. Treves, Demarks about certain first order linear PDE in two variables, Comm. Partial Differential Equations, 5 (1980), 381-425.  doi: 10.1080/0360530800882143.  Google Scholar

[20]

J. Sjöstrand, Note on a paper of F. Treves concerning Mizohata type operators, Duke Math. J., 47 (1980), 601-608.   Google Scholar

[21]

M. Yamamoto, On partially hypoelliptic operators, Osaka Math. J., 15 (1963), 233-247.   Google Scholar

show all references

References:
[1]

M. S. BaouendiC. H. Chang and F. Treves, Microlocal Hypo-Analytic and Extension of CR Functions, J. Differ. Geom., 18 (1983), 331-391.   Google Scholar

[2]

M. S. Baouendi and F. Treves, A local constancy principle for the solutions of certain overdetermined systems of first-order linear partial differential equations, in Advances in Mathematics Supplementary Studies, Academic Press, New York, (1981), 245-262.  Google Scholar

[3]

S. Berhanu, P. D. Cordaro and J. Hounie, An introduction to involutive structures, New mathematical monographs, University Press, (2008).  Google Scholar

[4]

P. D. Cordaro, Resolubilidade das Equações Diferenciais Parciais Lineares, Matemática Universitária, 14, (1992), 51-67. Google Scholar

[5]

P. D. Cordaro, Sistemas de Campos Vetoriais Complexos, Instituto de matemática pura e aplicada, (1986). Google Scholar

[6]

L. Garding and B. Malgrange, Opérateurs différentiels partiellement hypoelliptiques et partiellement elliptiques, Math. Scand., 9 (1961), 5-21.  doi: 10.7146/math.scand.a-10619.  Google Scholar

[7]

V. Grushin, A differential Equation without a solution, Mathematical Notes, 10 (1971), 499-501.   Google Scholar

[8]

N. Hanges, Almost mizohata operators, Trans. Amer. Math. Soc., 2 (1986), 663-675.  doi: 10.2307/2000030.  Google Scholar

[9]

G. Hoepfner and R. Medrado, Microlocal regularity for Mizohata type differential operators, J. Inst. Math. Jussieu, 19 (2020), 1185-1209.  doi: 10.1017/S1474748018000361.  Google Scholar

[10]

L. Hörmander, Differential equations without solutions, Math. Ann., 140 (1960), 169-173.  doi: 10.1007/BF01361142.  Google Scholar

[11]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Springer-Verlag, 1989. doi: 10.1007/978-3-642-61497-2.  Google Scholar

[12]

H. Lewy, An example of a smooth linear partial differential equation without solution, Ann. Math., 66 (1957), 155-158.  doi: 10.2307/1970121.  Google Scholar

[13]

S. Mizohata, Une remarque sur les opérateurs différentielshypoelliptiques et partiellement hypoelliptiques, J. Math. Kyoto Univ., 1-3 (1962), 411-423.  doi: 10.1215/kjm/1250525013.  Google Scholar

[14]

H. Ninomiya, A necessary and sufficient condition of local integrability, J. Math Kyoto Univ., 39-4 (1999), 685-696.  doi: 10.1215/kjm/1250517821.  Google Scholar

[15]

L. Nirenberg and F. Treves, On local solvability of linear partial differential equations, I. Necessary conditions, Commun. Pure Appl. Math, 23 (1970), 1-38.  doi: 10.1002/cpa.3160230102.  Google Scholar

[16]

L. Nirenberg and F. Treves, On local solvability of linear partial differential equations, II. Sufficient conditions, Commun. Pure Appl. Math, 23 (1970), 459-510.  doi: 10.1002/cpa.3160230314.  Google Scholar

[17]

L. Nirenberg and F. Treves, Solvability of a order linear partial differential equation, Commun. Pure Appl. Math, 16 (1963), 331-351.  doi: 10.1002/cpa.3160160308.  Google Scholar

[18]

L. Nunes, Resolubilidade Local Para Duas Classes de Campos de Vetores Suaves Complexos, Ph. D. thesis (in portuguese), UFSCar, 2016. Google Scholar

[19]

F. Treves, Demarks about certain first order linear PDE in two variables, Comm. Partial Differential Equations, 5 (1980), 381-425.  doi: 10.1080/0360530800882143.  Google Scholar

[20]

J. Sjöstrand, Note on a paper of F. Treves concerning Mizohata type operators, Duke Math. J., 47 (1980), 601-608.   Google Scholar

[21]

M. Yamamoto, On partially hypoelliptic operators, Osaka Math. J., 15 (1963), 233-247.   Google Scholar

Figure 1.  Integration domain complexification
[1]

Patrick Bonckaert, P. De Maesschalck. Gevrey and analytic local models for families of vector fields. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 377-400. doi: 10.3934/dcdsb.2008.10.377

[2]

Onur Alp İlhan. Solvability of some partial integral equations in Hilbert space. Communications on Pure & Applied Analysis, 2008, 7 (4) : 837-844. doi: 10.3934/cpaa.2008.7.837

[3]

Wei Xi Li, Chao Jiang Xu. Subellipticity of some complex vector fields related to the Witten Laplacian. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2709-2724. doi: 10.3934/cpaa.2021047

[4]

Vincent Naudot, Jiazhong Yang. Finite smooth normal forms and integrability of local families of vector fields. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 667-682. doi: 10.3934/dcdss.2010.3.667

[5]

Yuta Kugo, Motohiro Sobajima, Toshiyuki Suzuki, Tomomi Yokota, Kentarou Yoshii. Solvability of a class of complex Ginzburg-Landau equations in periodic Sobolev spaces. Conference Publications, 2015, 2015 (special) : 754-763. doi: 10.3934/proc.2015.0754

[6]

Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481

[7]

Jifeng Chu, Zhaosheng Feng, Ming Li. Periodic shadowing of vector fields. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3623-3638. doi: 10.3934/dcds.2016.36.3623

[8]

Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481

[9]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[10]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703

[11]

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053

[12]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515

[13]

Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032

[14]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167

[15]

Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345

[16]

Marcus A. Khuri. On the local solvability of Darboux's equation. Conference Publications, 2009, 2009 (Special) : 451-456. doi: 10.3934/proc.2009.2009.451

[17]

José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213

[18]

BronisŁaw Jakubczyk, Wojciech Kryński. Vector fields with distributions and invariants of ODEs. Journal of Geometric Mechanics, 2013, 5 (1) : 85-129. doi: 10.3934/jgm.2013.5.85

[19]

Davi Obata. Symmetries of vector fields: The diffeomorphism centralizer. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4943-4957. doi: 10.3934/dcds.2021063

[20]

A. V. Bobylev, Vladimir Dorodnitsyn. Symmetries of evolution equations with non-local operators and applications to the Boltzmann equation. Discrete & Continuous Dynamical Systems, 2009, 24 (1) : 35-57. doi: 10.3934/dcds.2009.24.35

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (62)
  • HTML views (112)
  • Cited by (0)

Other articles
by authors

[Back to Top]